Does Blender need Volume for SSS?
To work properly, yes, it does require volume. The Subsurface Scattering emulates scattering within the surface and this includes light that has travelled into the surface and has been scattered within the volume and back out of the surface, as well as light that has passed into the volume from a different surface and scattered back to the surface. If the mesh is a flat surface without volume then there is no volume to emulate and so no light can scatter through from the 'back' surface. However, the light from the top surface can still scatter so the shader will operate, just will be darker - without any effect from light striking the "back" surface (or sides).

To demonstrate I created a scene with a Subsurface Scatter shader on a plane illuminated by spotlights (one tinged red, the other blue) above and below. 

With a Solidify modifer :



Without solidify :